Thanks to their quick placement and high flexibility, unmanned aerial vehicles (UAVs) can be very useful in the current and future wireless communication systems. With a growing number of smart devices and infrastructure-free communication networks, it is necessary to legitimately monitor these networks to prevent crimes. In this paper, a novel framework is proposed to exploit the flexibility of the UAV for legitimate monitoring via joint trajectory design and energy management. The system includes a suspicious transmission link with a terrestrial transmitter and a terrestrial receiver, and a UAV to monitor the suspicious link. The UAV can adjust its positions and send jamming signal to the suspicious receiver to ensure successful eavesdropping. Based on this model, we first develop an approach to minimize the overall jamming energy consumption of the UAV. Building on a judicious (re-)formulation, an alternating optimization approach is developed to compute a locally optimal solution in polynomial time. Furthermore, we model and include the propulsion power to minimize the overall energy consumption of the UAV. Leveraging the successive convex approximation method, an effective iterative approach is developed to find a feasible solution fulfilling the Karush-Kuhn-Tucker (KKT) conditions. Extensive numerical results are provided to verify the merits of the proposed schemes.